(Update May 15, 2013: at this point, to my amazement, all of my questions have been answered, although more good answers to the rest would be most welcome. I have learned a great deal of neat stuff I should have known long before by asking these questions!)
A new version is now posted at the usual place (the Feb. 25, 2013 version). There are many small improvements and patches, but no important changes. Because I want to give you something new to look at, here is a newly added diagram of chapter dependencies. I was a little surprised by what it showed.
Diagram of Chapter Dependencies
The “to-do” list of things to be worked on is now at about 100 items. With the exception of formatting, figures, the bibliography, and the index, I continue to be very interested in hearing of any suggestions or corrections you might have, no matter how small. We are clearly nearing the endgame.
I now want to ask advice on a number of issues all at once. These are mostly small things, and are along the lines of “what is the best reference to point learners to on this topic”. Some of the questions are “unimportant”, in the sense that I doubt they will affect my exposition (although they may be important in some larger sense). For those questions relating to some particular part of the notes, I will give the section number. Please feel free to respond by email or in the comments. Here we go!
Peter Johnson strongly preferred using “fibre” instead of “fiber”. Does anyone else feel strongly?
1.4.1 Latex question: how do you get the \varprojlim subscript in the right place? (answer here)
1.6.12 (unimportant) Do right-exact functors always commute with colimits? (For example, commutes with direct sums, which is what we use, but that is easy to check directly.) (answer here)
5.4.M Can anyone get this exercise (that, basically, says that normality descends under finite field extensions)? I think it should be gettable, but not easy, but I’ve had clues that this is harder than I thought. I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including this one)
5.4.N is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s. Is there some good reference for this? (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case. And of course I don’t just want to know what is true; I’d like a reference for why it is true.) Added later: I should also have added, is there a well-loved reference that shows that the class group of is ? (answer here)
6.3.K A compact complex variety can have only one algebraic structure. What is a reference? (A number of sources mention this fact, but I want an actual proof.) On a related point, in 10.3: A variety over is proper if and only if it is compact in the “usual” topology. What is a reference? (answer here and here)
6.7 In this section, I mention the Schubert cell decomposition of the Grassmannian. The key idea is that any -dimensional subspace of (where is a field; and say , …, is the standard basis of ) has a canonical basis, where the first to appear in each basis element appears with coefficient 1, and that appears in no other basis element, and that special for that basis element is “to the right” of the of the previous one. Is there a standard name for this? (Normal form? Row-reduced echelon form?) Is there a good (fairly standard) reference for it? (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.) (answer here, although I’m also happy to get more references)
8.4.H Interesting fact: I almost wanted to say that effective Cartier divisors are the same as codimension 1 regular embeddings. But I could only show this in the locally Noetherian situation (or more generally, when the structure sheaf is coherent). The reason for the problem is that the definition of effective Cartier divisor is in terms of open subsets (for good reason), while the definition of regular embedding is in terms of stalks (for good reason), and getting from the latter to the former requires Nakayama. If you think I’m not giving the right definition of one of these two notions, please let me know! (see here for an interesting follow-up, thanks to Laurent Moret-Bailly)
9.1.7 Peter Johnson did not like my use of the phrase “open subfunctor” in 9.1.7. Is anyone else bothered? How seriously? (current plan after discussing with Peter: leave as is)
9.4.E Can anyone get this exercise (that, basically, says the product of integral varieties over an algebraically closed field is also an integral variety)? I think it should be gettable, but no one I know has gotten it (possibly because I haven’t asked it in homework sets). I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including from Gyujin Oh)
10.3.9 Is there an example of a non-smooth group variety over a field , i.e. a finite type reduced group scheme over that is not smooth? Translation: is there a group variety that is not an algebraic group? (answer: yes! example here)
11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least in is not uniruled. What is a good reference? (I know why it is true! As with many of these questions, I’d like to know where to point people to.) (answer here)
13.8 I mention Tate’s theory of non-archimedean analytic geometry. Is there a “right” source to point the interested reader (who is just starting out) to? (possible answer here)
19.9.B In (188.8.131.52), we have , and the discussion is away from characteristic . I want to say that the normalization factor is because of characteristic , but I couldn’t convince myself that this was true. Presumably it is. Is there a good reference? (Remark for comparison: one can also write in terms of : . Here the prime factors of are and ; but the reason for the is not characteristic .) (answer: yes, see here)
20.2.H Suppose is a complex elliptic curve. Then is always or . It is if there is a nontrivial endomorphism from to itself (i.e. not just multiplication by followed by translation); the additional class comes from the graph of this endomorphism. Is there a reference for this fact that I can/should direct learners to? (answer: yes, see here)
21.5.9 Is there a good reference for the Lefschetz principle? (Examples currently mentioned: Kodaira vanishing in characteristic 0; and non-jumping of hodge numbers in characteristic 0.) (good answer here)
21.7.8 (not needed) It is a nontrivial fact that irreducible smooth projective curves of
genus have finite automorphism groups. I know three arguments: using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points. I am surprised that this is so hard. (Note: I know that the idea can be quickly outlined to someone learning. But I want an easy complete rigorous proof. As long as I am asking, I also want someone to give me a Tesla Roadster.)
21.7.9 Smooth curves in positive characteristic can have way more than automorphisms. Is there a “best” reference?
28.1.L Is there a canonical reference for Tsen’s theorem, that any proper flat morphism to a curve, whose geometric fibers are isomorphic to is a Zariski -bundle? Follow-up question (posted March 5), in response to David Speyer’s comment here: Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a -bundle? (See David Speyer’s comment for a little more detail.) (possible answer to the first question here; answer to the second question here)
29.3.B I currently define node only in the case of a variety over an algebraically closed field, in which case I say that it is something formally isomorphic to . I gesture toward the definition in other cases. For example, if is not algebraically closed, I define it as , where is a quadratic with no repeated roots. I want to say that if is reducible, then this is said to be a split node, and otherwise it is a non-split node. I’d thought this was standard notation, but google suggests otherwise. Does anyone have strong feelings about this?
29.5 (This is a follow-up to discussion in the 27th post.) I am reluctant to introduce new terminology in a well-established field, but there is a notion that I think deserves a name. Suppose is a proper morphism. (For the technically-minded, it is likely that “finitely presented” should also be added, but I will play it safe, and not include this.) Then I want to say that is [something] if the natural map is an isomorphism. Not EinStein suggested the name -connected, and I quite like this — it suggests that this notion is even stronger than connected, and suggests in what way it is stronger. Another possibility is -isomorphic (which I suggested, but which I currently like less well). Opinions? (Are you offended by giving this a new name? Or do you like one of these suggestions? Or do you have another idea?)
30.3.4 Is there a canonical (“introductory”) reference for (which will require an introduction to derived categories)? (Brian Conrad’s book Grothendieck duality and base change perhaps?) (possible answer here)